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Mirrors > Home > MPE Home > Th. List > tnggrpr | Structured version Visualization version GIF version |
Description: If a structure equipped with a norm is a normed group, the structure itself must be a group. (Contributed by AV, 15-Oct-2021.) |
Ref | Expression |
---|---|
tngngp3.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
Ref | Expression |
---|---|
tnggrpr | ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngngp3.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
2 | eqid 2738 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | tngbas 23387 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝑇)) |
4 | eqidd 2739 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | eqid 2738 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | tngplusg 23388 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝑇)) |
7 | 6 | eqcomd 2744 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝑇) = (+g‘𝐺)) |
8 | 7 | oveqdr 7192 | . . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝑇)𝑦) = (𝑥(+g‘𝐺)𝑦)) |
9 | 3, 4, 8 | grppropd 18229 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑇 ∈ Grp ↔ 𝐺 ∈ Grp)) |
10 | 9 | biimpd 232 | . 2 ⊢ (𝑁 ∈ 𝑉 → (𝑇 ∈ Grp → 𝐺 ∈ Grp)) |
11 | ngpgrp 23345 | . 2 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp) | |
12 | 10, 11 | impel 509 | 1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 +gcplusg 16661 Grpcgrp 18212 NrmGrpcngp 23323 toNrmGrp ctng 23324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-plusg 16674 df-tset 16680 df-ds 16683 df-0g 16811 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-grp 18215 df-ngp 23329 df-tng 23330 |
This theorem is referenced by: tngngp3 23402 |
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